Application 02 · personal propulsion
The
Hoverboard.
Two lawful routes.
A thin coherent-matter disc on the underside drives a small modulation of the OPH coupling ν. The open question is which of the two lawful force routes the substrate can supply: nonzero device repair charge with a field tail, or explicit momentum transfer through the repair-field stress tensor.
χν canonical band
0.9320 – 1
Mueller-Osika · exact 1 − P/24 (presence branch)
Dimensional coupling q★
open
the coefficient that turns the χν band into a force · work in progress
Reactionless lift
excluded
no compact neutral device lifts itself (χν bounds paper)
Substrate ΔS_coh
open
the coherence contrast a real substrate can hold under load
Live simulation · illustration only
We don't push against gravity. We edit the substrate.
Weight, in OPH, is what observers agree about when a patch of coherent matter renegotiates its overlap with the rest of the network. Drive the underside of the deck into a matter-coherent state and the local ν interpolation tilts by δν along z. The gravitational bookkeeping for that patch changes.
The slider below sweeps a scalar response amplitude, bounded above by the canonical band 0.9320 ≤ χν ≤ 1 and below by whatever coherence contrast ΔScoh the skin can hold. It is not a force law. No formula proportional only to g²·A·χν·ΔS/(4πG) is a force theorem for a closed compact device — the χν bounds paper settles that. Treat the widget as an illustration of the continuation law δν = χν · Scoh, not as a rider becoming weightless.
·χν band·reactionless lift · excluded·q★ (force law)·substrate ΔS_coh under load
Interactive · χν drive sweep
Drag the drive. Watch effective mass collapse.
True mass
6.4 kg
deck + electronics
Effective mass
6.40 kg
m − Fl / g
χν lift (Fl)
0.0 N
weight Fg = 62.8 N
Net Fz
-62.8 N
pinned down
Illustration of the continuation law δν = χν · Scoh. The displayed "effective mass" tracks the scalar amplitude the slider sweeps; it is not a derived force on a closed compact device. The dimensional coupling that would convert the χν band into a real body force (q★) is work in progress.
Coherent-matter skin
The underside of the disc is driven into a matter-coherent, overlap-consistent state.
Local ν shift
The coherent-matter scalar S couples through χν. ν gets nudged by δν, asymmetrically along z.
Body force
The dark-sector source equation turns ∇·[(ν−1)g_b] into a static vertical force on the disc.
Force conversion is open
Effective weight changes only if the device carries net repair charge in an external gradient, or transfers momentum through the repair-field stress tensor. The dimensional coupling q★ that closes this step is work in progress.
How you actually lift it
Two regimes, one piece of hardware.
The deck is, mechanically, an array of resonant metal radiators driven by piezo transducers and a cheap microcontroller. The same hardware sits on two rungs of the substrate ladder. The lower rung works today on textbook physics. The upper rung is what the χν band is for.
RUNG 01 · χν = 0 · works today
Near-field acoustic levitation
Four 20″ bronze cymbals are mounted bell-down at the corners of the deck. Twelve piezo transducers drilled through them are driven by a phase-locked broadband multichord signal in the 200 Hz – 2 kHz band, tuned to the cymbals' eigenmodes. A standing wave forms in the thin air cavity between the deck and the ground; the time-averaged radiation pressure lifts the deck. Cavity-Q does the rest.
- Substrate · 4× 20″ bronze cymbals (~€200 used)
- Drive · 12 piezos · phase-locked multichord
- Brain · €12 microcontroller running the chord pattern
- Stokes floor · ≈ 90 W for 150 kg in ground effect
- Realistic draw · 100 – 300 W depending on floor & gap
Most of the coherent energy lives in the air column, not the metal — the air cavity stores ~20× more than the substrate. The cymbal's only job is to keep that air state alive.
Limits · works in atmosphere, in ground effect only. No vacuum operation, no free altitude.
RUNG 02 · χν > 0 · scalar response · open engineering·force conversion open
Vertex-sharing strength g = χν
The same coherent vibration the cymbals already maintain attaches to the spacetime lattice as a fourth port at each three-way vertex. The dominant element of that 4×4 scattering matrix is the vertex-sharing strength g, which this site also calls χν. Per vertex it is bounded by unitarity + 3-fold symmetry + time reversal at 0 ≤ g ≤ 1/√3 ≈ 0.577. Per substrate, g is the coherent sum over every vertex the skin holds in lock. That establishes the response channel; it does not by itself lift a closed compact device.
- Bias the lattice scattering, not the air
- Asymmetric coherent vibration on the deck underside
- Required substrate contrast ΔS_coh ≈ 10⁻⁸ under load
- Same hardware — different substrate, different firmware
- Force conversion (q★) has to close before a compact device lifts itself
Implementation note · anywhere the legacy bench notation says χν, read g. They are the same number under two names.
What Rung 02 buys you today · a lawful scalar response channel to measure ΔScoh against, on a torsion pendulum. Free-altitude flight is not on this rung — the χν bounds paper excludes reactionless lift from a closed compact device.
The weekend build is Rung 01. You can fly it on a polished floor this month for the cost of a used cymbal set, a fistful of piezos, and an ESP32. Rung 02 is the same airframe with a substrate swap — the χν band tells you exactly what coherence contrast that swap has to deliver. See the two source papers below for the full electronics, safety stop, and substrate ladder.
Technical detail
Why it actually works.
OPH derives gravity as the Jacobson-style thermodynamic consequence of overlap-consistency on the prime geometric subnet. The dark-sector remainder is sourced by an interpolation function ν(x). On Earth ν → 1 and the anomaly vanishes. The χν continuation lets a coherent-matter scalar locally shift ν by a tiny δν.
On the declared quotient-edge branch, χν is not a free parameter. The Mueller-Osika collar lemmas pin the canonical coefficient to the theorem-grade band 0.9320 ≤ χνcan ≤ 1, with exact presence-branch value 1 − P/24 = 0.9320429912748350…. The engineering question is the coherence contrast ΔScoh the substrate can hold, and the dimensional coupling q★ that would convert the χν band into a body force. Both are open. See Theoretical Bounds on χν (PDF).
Dark-sector source equation · OPH-canonical
Non-baryonic gravitating sector. See Dark Matter and Recovering Relativity papers.
χν susceptibility · canonical band (Mueller-Osika r1577)
Theorem on the declared quotient-edge collar branch. exp(−P/24) is excluded — the χν bounds paper states it is the supremum of the family, attained by no finite regulator.
Continuation law · scalar response
Establishes a lawful scalar response channel. Converting δν to a body force on a compact device requires the dimensional coupling q★, which is work in progress.
What the χν bounds paper excludes
Reactionless lift from a compact neutral contrast is excluded. A static, closed, repair-neutral device cannot lift itself. Momentum has to be tracked through repair charge in an external gradient or through the stress ledger.
Coherence contrast · derived on this page·derived here
What the χν bound would demand of the substrate if a force law existed.
Setting the canonical band at 0.9320 ≤ χνcan ≤ 1 and solving ΔScoh = Δν / χνcan for a target Δν gives the substrate contrast the χν disc would need to hold. These numbers are derived on this page, not lifted from a paper table, and they assume a force conversion the χν bounds paper does not yet supply (see q★, work in progress).
| Device case | Σ (kg/m²) | f | Δν required | ΔS_coh^can required |
|---|---|---|---|---|
| Light room platform | 50 | 1.0 | 4.28 × 10⁻⁹ | 4.4 – 4.6 × 10⁻⁹ |
| Room-scale platform | 100 | 1.0 | 8.55 × 10⁻⁹ | 8.7 – 9.2 × 10⁻⁹ |
| Heavy room platform | 250 | 1.0 | 2.14 × 10⁻⁸ | 2.2 – 2.3 × 10⁻⁸ |
| Hoverboard footprint (rider + board) | 200 – 300 | 1.0 | 1.7 – 2.6 × 10⁻⁸ | 1.8 – 2.7 × 10⁻⁸ |
| Compact hoverboard footprint | 600 | 1.0 | 5.13 × 10⁻⁸ | 5.2 – 5.5 × 10⁻⁸ |
| Ten percent assist | 100 | 0.1 | 8.55 × 10⁻¹⁰ | 8.7 – 9.2 × 10⁻¹⁰ |
The coefficient is in the useful mathematical range for hoverboard-class experiments. The remaining engineering question is whether a real substrate can produce and hold that vertical scalar contrast under load, while keeping ambient ordinary matter from generating the same scalar accidentally.
Bench specs · target envelope·target envelope
Where a χν-drive hoverboard would sit next to existing vehicles.
The OMEGA row is a target envelope, not a measured device. It assumes q★ closes and the substrate holds the required ΔScoh; both are open.
| Vehicle | True mass | Active disc area | Lift needed | Range |
|---|---|---|---|---|
| Hoverboard (OMEGA · target) | 78 kg | 0.18 m² | 0.77 kN | battery-limited |
| Electric scooter | 120 kg | — | — | 40 km |
| Quadcopter (human-rated) | 350 kg | — | 3.4 kN | 20 min |
| Helicopter (R22) | 620 kg | — | 6.1 kN | 350 km |
A χν-drive vehicle would carry no rotor, no exhaust, no propellant tank. It would still owe momentum to the substrate through the repair-field stress tensor; that ledger has to balance, which is why reactionless lift on a closed compact device is excluded.
The paper trail
The 5 OPH papers this page leans on.
The χν disc is an extension hypothesis by Alex Osika on top of the OPH canon. Each link below is a paper the hoverboard argument depends on directly, with the specific connection spelled out. The full full 12-paper corpus lives on the hub.
Theoretical Bounds on χν in Observer-Patch Holography
Mueller & Osika, r1577. On the declared quotient-edge collar branch, the canonical susceptibility is pinned to 0.9320 ≤ χν^can ≤ 1, with exact presence-branch value 1 − P/24 = 0.9320429912748350…. χν is a bounded design parameter; the open question is the coherence-contrast ΔS_coh a substrate can hold under load.
Connection to this page
The decisive paper for this page. On the declared quotient-edge collar branch it pins the canonical χν to 0.9320 ≤ χν^can ≤ 1, with exact presence-branch value 1 − P/24 = 0.9320429912748350…. It also proves the negative results this page respects: reactionless lift from a compact neutral contrast is excluded, and a static, closed, repair-neutral device cannot lift itself.
Observers Are All You Need
Foundational paper. Derives the patch-network consensus framework from finite-observer overlap-consistency.
Connection to this page
Defines the patch-network consensus the χν disc has to remain compatible with. If a hoverboard violated cap-consistency, this paper is what it would break.
Recovering Relativity and Standard-Model Structure from Observer-Overlap Consistency
Derives the Einstein equation in Jacobson form and Standard-Model gauge structure from cap-consistency on the prime geometric subnet.
Connection to this page
Derives the Einstein equation from overlap consistency. Sets the baseline: gravity is already an information-theoretic effect, so locally modulating ν is a legal move, not a new force.
OPH Dark Matter
Introduces ν_OPH as the interpolation function that sources the information-defect remainder. The ν coupling this whole page modulates.
Connection to this page
Introduces the ν_OPH coupling that the coherent-matter disc is supposed to bend. Every number in the spec table is a perturbation of the function this paper defines.
Screen Microphysics and Observer Synchronization
Mueller, r1577. Owns the finite carrier and its public interfaces on the Echosahedral branch: twelve-port oriented boundary with incidence (V,E,F)=(12,30,20), the federation screen, the support screen, and the presentation-invariance theorem for the observer-visible carrier signature.
Connection to this page
Specifies how holographic-screen microphysics enforces synchronization between observers. The coherent-matter skin is an engineered version of the screen this paper describes.
Claim boundary · three tiers
Tier A · recovered OPH core: patch carriers, mismatch-lowering repair, record algebras, checkpoint continuation, Jacobson-type Einstein branch, and the canonical dark-sector scalar channel ρA = −(1/4πG)∇·[(νOPH−1)gb]. Earth-surface null at first order.
Tier B · continuation law: declares the coherent-matter response δν = χν · Scohin both canonical and engineering charts. Existence of the response channel; χν unfixed.
Tier C · branch theorem (Mueller-Osika r1577): on the declared quotient-edge collar branch, 0.9320 ≤ χνcan ≤ 1 with exact presence-branch value 1 − P/24 = 0.9320429912748350…. Zero is excluded on that branch; exp(−P/24) is the family's supremum and is itself excluded.
Not claimed: built hardware, a working force conversion (q★ is open), a real substrate that holds the required vertical ΔScohunder load, or reactionless lift from a compact device (excluded by the χν bounds paper). The first receipt is a measurement of the coherence contrast on a controlled torsion-pendulum protocol. Work in progress.